aa r X i v : . [ h e p - ph ] J a n TU–832
Family Gauge Symmetry and Koide’s Mass Formula
Yukinari Sumino
Department of Physics, Tohoku University, Sendai, 980–8578 Japan (Dated: October 24, 2018)Koide’s mass formula is an empirical relation among the charged lepton masses which holds witha striking precision. We propose a mechanism for cancelling the QED correction to Koide’s formula.This is discussed in an effective theory with U (3) family gauge symmetry and a scenario in whichthis symmetry is unified with SU (2) L symmetry at 10 –10 TeV scale.
PACS numbers: 11.30.Hv,12.10.Kt,12.15.Ff,13.20.Eb
Among various properties of elementary particles, thespectra of the quarks and leptons exhibit unique pat-terns, and their origin still remains as a profound mys-tery. Koide’s mass formula is an empirical relation amongthe charged lepton masses given by [1] √ m e + √ m µ + √ m τ √ m e + m µ + m τ = r , (1)which holds with a striking precision. In fact, substitut-ing the present experimental values of the charged leptonmasses [2], the formula is valid within the present experi-mental accuracies. The relative experimental error of theleft-hand side of eq. (1) is of order 10 − .Given the remarkable accuracy with which eq. (1)holds, there have been many speculations as to existenceof some physical origin behind this mass formula [1, 3].Despite the attempts to find its origin, so far no realis-tic model or mechanism has been found which predictsKoide’s mass formula within the required accuracy. Themost serious problem one faces in finding a realistic modelor mechanism is caused by the QED radiative correction.Even if one postulates some mechanism at a high energyscale that leads to this mass relation, the charged leptonmasses receive the 1-loop QED radiative correction givenby m pole i = (cid:20) απ (cid:26)
34 log (cid:18) µ ¯ m i ( µ ) (cid:19) + 1 (cid:27)(cid:21) ¯ m i ( µ ) , (2)where ¯ m ( µ ) and m pole denote the running mass defined inthe modified–minimal–subtraction scheme (MS scheme)and the pole mass, respectively; µ represents the renor-malization scale. It is the pole mass that is measuredin experiments. Suppose ¯ m i ( µ ) (or the correspondingYukawa couplings ¯ y i ( µ )) satisfy the relation (1) at scale µ ≫ M W . Then m pole i do not satisfy the same relation[5]: Eq. (1) is corrected by approximately 0.1%, whichis 120 times larger than the present experimental error.Note that this correction originates only from the term − α/ (4 π ) × ¯ m i log( ¯ m i ) of eq. (2), since the other terms,which are of the form const . × ¯ m i , do not affect the rela-tion (1). We also note that log( ¯ m i ) results from the factthat ¯ m i plays a role of an infrared cut–off in the loopintegral.The 1–loop weak correction is of the form const . × ¯ m i in the leading order of ¯ m i /M W expansion; the leading non–trivial correction is O ( G F ¯ m i /π ) whose effect canbe safely neglected. Other known radiative correctionsare also negligible.Thus, if there is indeed a physical origin to Koide’smass formula at a high energy scale, we need to accountfor a correction to the relation (1) that cancels the QEDcorrection. Since such a correction is absent up to thescale of O ( M W ) to our present knowledge, it must orig-inate from a higher scale. Then, there is a difficulty inexplaining why the size of such a correction should coin-cide accurately with the size of the QED correction whicharises from much lower scales. Up to date, no mechanismhas been proposed to solve this problem.Among various existing models which attempt to ex-plain origins of Koide’s mass formula, we find a class ofmodels particularly attractive [6]. These are the modelswhich predict the mass matrix of the charged leptons tobe proportional to the square of the vacuum expectationvalue (VEV) of a scalar field (we denote it as Φ) writtenin a 3–by–3 matrix form: M ℓ ∝ h Φ ih Φ i . (3)Thus, ( √ m e , √ m µ , √ m τ ) is proportional to the diagonalelements of h Φ i in the basis where it is diagonal. TheVEV h Φ i is determined by minimizing the potential ofscalar fields in each model. Hence, the origin of Koide’sformula is attributed to the specific form of the potentialwhich realizes this relation in the vacuum configuration.Up to now, no model is complete with respect to sym-metry: Every model requires either absence or strongsuppression of some of the terms in the potential (whichare allowed by the symmetry of that model), withoutjustification.In this paper, we propose a possible mechanism forcancelling the QED correction to Koide’s mass formula,in the context of a theory with family (horizontal) gaugesymmetry. In our study we adopt the mechanism eq. (3)for generating the charged lepton masses at tree level, dueto the following reasons. First, models with this mecha-nism are suited for perturbative analyses. Secondly, sinceΦ is renormalized multiplicatively, the structure of radia-tive corrections is simple.We consider an effective theory with family gauge sym-metry which is valid up to some cut–off scale denoted byΛ ( ≫ M W ). Within such an effective theory, the chargedlepton masses may be induced by a higher–dimensionaloperator O = κ ( µ )Λ ¯ ψ Li Φ ik Φ kj ϕ e Rj . (4)Here, ψ Li = ( ν Li , e Li ) T denotes the left–handed lepton SU (2) L doublet of the i –th generation; e Rj denotes theright-handed charged lepton of the j –th generation; ϕ de-notes the Higgs doublet field; Φ is a 9–component scalarfield and is singlet under the Standard Model (SM) gaugegroup. We suppressed all the indices except for the gen-eration (family) indices i, j, k = 1 , ,
3. (Summation overrepeated indices is understood throughout the paper.)The dimensionless Wilson coefficient of this operator isdenoted as κ ( µ ). Once Φ acquires a VEV, the operator O will effectively be rendered to the Yukawa interactionsof the SM; after the Higgs field also acquires a VEV, h ϕ i = (0 , v ew / √ T with v ew ≈
250 GeV, the operatorwill induce the charged–lepton mass matrix of the formeq. (3) at tree level: M tree ℓ = κ v ew √ h Φ ih Φ i . (5)We consider radiative corrections to the above massmatrix by the family gauge interaction. First we con-sider the case, in which the gauge group is SU (3) andboth ψ L and e R are assigned to the (fundamental rep-resentation) of this symmetry group. With this choiceof representation, however, Koide’s formula is subject toa severe radiative correction unless the family gauge in-teraction is strongly suppressed. In fact, the 1–loop cor-rection by the family gauge bosons induces an effectiveoperator O ′ ∼ α F π × κ ¯ ψ Li ϕ e Ri × h Φ i jk h Φ i kj Λ , (6)hence corrections universal to all the charged–leptonmasses, ( δm e , δm µ , δm τ ) ∝ (1 , , ψ Li ϕ e Ri is not prohibited by symmetry. Here, α F = g F / (4 π ) de-notes the gauge coupling constant of the family gauge in-teraction. As noted above, corrections which are propor-tional to individual masses do not affect Koide’s formula;oppositely, the universal correction violates Koide’s for-mula strongly. In order that the correction to Koide’s for-mula cancel the QED correction, a naive estimate showsthat α F /π should be order 10 − , provided that the cut–off Λ is not too large and that the above operator O ′ isabsent at tree level. If O ′ exists at tree level, there shouldbe a fine tuning between the tree and 1–loop contribu-tions. The situation is similar if the family symmetry is O (3) and both ψ L and e R are in the . In these cases[7] we were unable to find any sensible reasoning for thecancellation between the QED correction and the correc-tion induced by family gauge interaction, other than toregard the cancellation as just a pure coincidence. Hence,we will not investigate these choices of representation fur-ther. In the case that ψ L is assigned to and e R to ¯ (or viceversa ) of U (3) family gauge group, (i) the dimension–4operator ¯ ψ Li ϕ e Ri is prohibited by symmetry, and hencecorrections universal to all the three masses do not ap-pear; and (ii) marked resemblance of the radiative cor-rection to the QED correction follows. We show thesepoints explicitly in a specific setup.We denote the generators for the fundamental repre-sentation of U (3) by T α (0 ≤ α ≤ (cid:0) T α T β (cid:1) = 12 δ αβ ; T α = T α † . (7) T = √ is the generator of U (1), while T a (1 ≤ a ≤ SU (3).We assign ψ L to the representation ( , stands for the SU (3) representation and 1 for the U (1)charge, while e R is assigned to (¯ , − U (3), the9–component field Φ transforms as three ( , ψ L → U ψ L , e R → U ∗ e R , Φ → U Φ (8)with U = exp ( iθ α T α ).We assume that the charged–lepton mass matrix is in-duced by a higher–dimensional operator O ( ℓ ) similar to O in eq. (4). We further assume that h Φ i can be broughtto a diagonal form in an appropriate basis. Thus, in thisbasis O ( ℓ ) , after Φ and ϕ acquire VEVs, turns to thelepton mass terms as O ( ℓ ) → κ ( ℓ ) ( µ ) v ew √ ¯ ψ L Φ d ( µ ) e R , (9)whereΦ d ( µ ) = v ( µ ) 0 00 v ( µ ) 00 0 v ( µ ) , v i ( µ ) > . (10)When all v i are different, U (3) symmetry is completelybroken by h Φ i = Φ d , and the spectrum of the U (3) gaugebosons is determined by Φ d .Note that the operator O in eq. (4) is not invariantunder the U (3) transformations eq. (8). As an exampleof O ( ℓ ) , one may consider O ( ℓ )1 = κ ( ℓ ) ( µ )Λ ¯ ψ L Φ Φ T ϕ e R . (11)It is invariant under a larger symmetry U (3) × O (3), un-der which Φ transforms as Φ → U Φ O T ( O O T = ). Inthis case, we need to assume, for instance, that the O (3)symmetry is gauged and spontaneously broken at a highenergy scale before the breakdown of the U (3) symme-try, in order to eliminate Nambu–Goldstone bosons andto suppress mixing of the U (3) and O (3) gauge bosons.In any case, the properties of O ( ℓ ) given by eqs. (9) and(10) are sufficient for computing the radiative correctionby the U (3) gauge bosons to the mass matrix, withoutan explicit form of O ( ℓ ) .We take the U (1) and SU (3) gauge coupling constantsto be the same: α U (1) = α SU (3) = α F . (12)We compute the radiative correction in Landau gauge,which is known to be convenient for computations intheories with spontaneous symmetry breaking. Throughstandard computation, one obtains δm pole i = − α F π (cid:20) log (cid:18) µ v i ( µ ) (cid:19) + c (cid:21) m i ( µ ) , (13) m i ( µ ) = κ ( ℓ ) ( µ ) v ew √ v i ( µ ) . (14)Here, c is a constant independent of i . The Wilson co-efficient κ ( ℓ ) ( µ ) is defined in MS scheme. v i ( µ ) are de-fined as follows: The VEV of Φ at renormlization scale µ ,Φ d ( µ ) = h Φ( µ ) i given by eq. (10), is determined by min-imizing the 1–loop effective potential in Landau gauge(although we do not discuss the explicit form of the ef-fective potential); Φ is renormalized in MS scheme. Weignored terms suppressed by m i /v j ( ≪
1) in the aboveexpression. Note that the pole mass is renormalization–group invariant and gauge independent. Therefore, theabove expression is rendered gauge independent if we ex-press v i ( µ ) in terms of gauge independent parameters,such as coupling constants defined in on–shell scheme.The form of the radiative correction given by eqs. (13)and (14) is constrained by symmetries and their breakingpatterns. As the diagonal elements of the VEV, v >v > v >
0, are successively turned on, gauge symmetryis broken according to the pattern: U (3) → U (2) → U (1) → nothing . (15)At each stage, the gauge bosons corresponding to thebroken generators acquire masses and decouple. Further-more, the vacuum Φ d and the family gauge interactionrespect a global U (1) symmetry generated by ψ L → V ψ L , e R → V ∗ e R , Φ d → V Φ d V ∗ , (16)with V = e iφ e iφ
00 0 e iφ ; φ i ∈ R . (17)Although O ( ℓ ) after symmetry breakdown, eq. (9), is notinvariant under this transformation, the variation can beabsorbed into a redefinition of v i . As a result, the leptonmass matrix has a following transformation property: M ℓ (cid:12)(cid:12)(cid:12)(cid:12) v i → v i exp( iφ i ) = V M ℓ V ∗ . (18)This is satisfied including the 1–loop radiative correction.The symmetry breaking pattern eq. (15) and the above transformation property constrain the form of the radia-tive correction to δm pole i ∝ v i [log( | v i | ) + const . ], wherethe constant is independent of i . Note that | v i | in theargument of logarithm originates from the gauge bosonmasses, which are invariant under v i → v i exp( iφ i ).The universality of the SU (3) and U (1) gauge cou-plings eq. (12) is necessary to guarantee the above sym-metry breaking pattern eq. (15). One may worry aboutvalidity of the assumption for the universality, since thetwo couplings are renormalized differently in general.The universality can be ensured approximately if thesetwo symmetry groups are embedded into a simple groupdown to a scale close to the relevant scale. There are morethan one ways to achieve this. A simplest way would beto embed SU (3) × U (1) into SU (4). It is easy to ver-ify that the of SU (4) decomposes into ( , − ) ⊕ ( , )under SU (3) × U (1). Hence, the (second-rank anti-symmetric representation) and ¯ of SU (4), respectively,include (¯ , −
1) and ( , m i ( µ ) replaced by m i ( µ ). Recall that correctionsof the form const . × m i do not affect Koide’s formula.Noting log v i = log m i + const., one observes that ifa relation between the QED and family gauge couplingconstants α = 14 α F (19)is satisfied, the 1–loop radiative correction induced byfamily gauge interaction cancels the 1–loop QED correc-tion to Koide’s mass formula.Suppose the relation (19) is satisfied. Then m pole i ∝ v i ( µ ) (20)holds with a good accuracy. This is valid for any valueof µ . This means, if v i ( µ ) satisfy v ( µ ) + v ( µ ) + v ( µ ) p v ( µ ) + v ( µ ) + v ( µ ) = r
32 (21)at some scale µ , Koide’s formula is satisfied at any scale µ . This is a consequence of the fact that Φ is multiplica-tively renormalized. Generally, the form of the effectivepotential varies with scale µ . If the relation (21) is real-ized at some scale as a consequence of a specific natureof the effective potential (in Landau gauge), the same re-lation holds automatically at any scale. Although thesestatements are formally true, physically one should con-sider scales only above the family gauge boson masses,since decoupling of the gauge bosons is not encoded inMS scheme. An interesting possibility is to use eq. (20)to relate the charged lepton pole masses with the VEVat the cut–off scale, i.e. µ = Λ, which sets a boundary(initial) condition of the effective theory.The advantages of choosing Landau gauge in our com-putation are two folds: (1) The computation of the 1–loop effective potential becomes particularly simple (aswell known in computations of the effective potential invarious models); in particular there is no O ( α F ) cor-rection to the effective potential. (2) The lepton wave–function renormalization is finite; as a consequence, the p µ γ µ part of the lepton self–energy is independent of gen-eration. Due to the former property, there is no O ( α F )correction to the relation eq. (21) if it is satisfied attree level. Due to the latter property, the correction toKoide’s formula is determined by renormalization of O ( ℓ ) alone, and a simple relation to h Φ( µ ) i follows.Let us comment on gauge dependence of our predic-tion. If we take another gauge and express the radiativecorrection δm pole i in terms of h Φ( µ ) i , the coefficient oflog( µ / h Φ i ) changes, and other non–trivial flavor de-pendent corrections are induced. Suppose the relationeq. (21) is satisfied at tree level [11]. The VEV h Φ i in an-other gauge receives an O ( α F ) correction, which inducesa correction to eq. (21) at O ( α F ). These additional cor-rections to δm pole i at O ( α F ) should cancel altogether ifthey are reexpressed in terms of the tree–level v i ’s whichsatisfy eq. (21), since the O ( α F ) correction to the rela-tion (21) vanishes in Landau gauge. General analyses ongauge dependence of the effective potential may be foundin [9].Now we speculate on a possible scenario how the rela-tion (19) may be satisfied. The scale of α is determinedby the charged lepton masses, while the scale of α F is de-termined by the family gauge boson masses, which shouldbe much higher than the electroweak scale. Since the rel-evant scales of the two couplings are very different, weare unable to avoid assuming some accidental factor (orparameter tuning) to achieve this condition. Instead weseek for an indirect evidence which indicates such an ac-cident has occurred in Nature. The relation (19) showsthat the value of α F is close to that of the weak gaugecoupling constant α W , since sin θ W ( M W ) is close to 1 / α W ( µ ) approximates α ( m τ ) atscale µ ∼ –10 TeV. Hence, if the electroweak SU (2) L gauge group and the U (3) family gauge group are uni- fied around this scale, naively we expect that α ≈ α F is satisfied. Since α W runs relatively slowly in the SM,even if the unification scale is varied within a factor of 3,Koide’s mass formula is satisfied within the present ex-perimental accuracy. This shows the level of parametertuning required in this scenario.We assume that anomalies introduced by the cou-plings of fermions to family gauge bosons are cancelled,which requires existence of fermions other than the SMfermions. Furthermore, we assume that all the additionalfermions acquire masses of order h Φ i or larger after thespontaneous breakdown of U (3), so that they decouplefrom the SM sector at and below the electroweak scale.A characteristic prediction of the present scenario isthe existence of lepton flavor violating processes at 10 –10 TeV scale. For instance, assuming that the down–type quarks are in the same representation of U (3) asthe charged leptons, and that the mass matrices ofthe charged leptons and down–type quarks are simul-taneously diagonalized in an appropriate basis, we findΓ( K L → eµ ) ≈ m µ m K L f K / (16 πv ). Comparing to thepresent experimental bound Br( K L → eµ ) < . × − [2], we obtain a limit v > ∼ × TeV. Naively this limitmay be marginally in conflict with the estimated unifica-tion scale in the above scenario. This depends, however,rather heavily on our assumptions on the quark sector. Inthe case that there exist additional factors in the quarksector which suppress the decay width by a few ordersof magnitude, we may expect a signal for K L → eµ notfar beyond the present experimental reach. Predictionsconcerning purely leptonic processes are less model de-pendent, but expected event rates are far below presentexperimental sensitivities.Models, which predict a realistic charged lepton spec-trum incorporating the mechanism proposed in this pa-per, will be discussed elsewhere [10].The author is grateful to K. Tobe for discussion. Thiswork is supported in part by Grant-in-Aid for scientificresearch No. 17540228 from MEXT, Japan. [1] Y. Koide, Nuovo Cim. A (1982) 411 [Erratum-ibid. A (1983) 327].[2] C. Amsler et al. [Particle Data Group], Phys. Lett. B (2008) 1.[3] Earlier works are Y. Koide, Phys. Rev. D (1983) 252;R. Foot, arXiv:hep-ph/9402242; S. Esposito and P. San-torelli, Mod. Phys. Lett. A (1995) 3077. See also areview [4] and references therein.[4] Y. Koide, arXiv:hep-ph/0506247.[5] N. Li and B. Q. Ma, Phys. Rev. D (2006) 013009;Z. z. Xing and H. Zhang, Phys. Lett. B (2006) 107.[6] Earlier works are Y. Koide, Mod. Phys. Lett. A (1990)2319; Y. Koide and H. Fusaoka, Z. Phys. C (1996)459; Y. Koide and M. Tanimoto, Z. Phys. C (1996)333. See also a review [4] and references therein.[7] There are a large number of papers on the fermion flavor structure based on SU (3) or SO (3) family symmetry. See,for instance, S. F. King, JHEP (2005) 105; I. deMedeiros Varzielas and G. G. Ross, Nucl. Phys. B (2006) 31; S. Antusch, S. F. King and M. Malinsky, JHEP (2008) 068, and references therein.[8] O. M. Del Cima, D. H. T. Franco and O. Piguet, Nucl.Phys. B (1999) 813.[9] Earlier works are L. Dolan and R. Jackiw, Phys. Rev. D , 2904 (1974); N. K. Nielsen, Nucl. Phys. B , 173(1975); R. Fukuda and T. Kugo, Phys. Rev. D , 3469(1976). See also [8] and references therein.[10] Y. Sumino, [arXiv:0812.2103 [hep-ph]]; Y. Sumino, inpreparation.[11] To simplify the argument we consider only those gaugesin which tree–level vacuum configuration is gauge inde-pendent, such as the class of gauges considered in [8]., 3469(1976). See also [8] and references therein.[10] Y. Sumino, [arXiv:0812.2103 [hep-ph]]; Y. Sumino, inpreparation.[11] To simplify the argument we consider only those gaugesin which tree–level vacuum configuration is gauge inde-pendent, such as the class of gauges considered in [8].